This paper presents a new algorithm for constrained intensity-modulated radiotherapy (IMRT) planning, made tractable by a dimensionality reduction using a set of plans obtained by fast, unconstrained optimizations. The main result is to reduce planning time by an order of magnitude, producing viable five field prostate IMRT plans in about 5 min. Broadly, the algorithm has three steps. First, we solve a series of independent unconstrained minimization problems based on standard penalty-based objective functions, 'probing' the space of reasonable beamlet intensities. Next, we apply principal component analysis (PCA) to this set of plans, revealing that the high-dimensional intensity space can be spanned by only a few basis vectors. Finally, we parameterize an IMRT plan as a linear combination of these few basis vectors, enabling the fast solution of a constrained optimization problem for the desired intensities. We describe a simple iterative process for handling the dose–volume constraints that are typically required for clinical evaluation, and demonstrate that the resulting plans meet all clinical constraints based on an approximate dose calculation algorithm.